In this paper, the nonlocal elasticity theory has been incorporated into classical 1D-rod model to capture unique features of the rod like structures at Nanoscale, which are considered as ultra-thin structures, under the umbrella of continuum mechanics theory. The strong effect of the nanoscale has been obtained which leads to substantially different wave behaviors of nanoscale-rods from those of macroscopic rods. Nonlocal bar model is developed for nanorods. The analysis shows that the wave characteristics are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The studies also show that the nonlocal scale parameter introduces certain band gap region in axial wave mode where no wave propagation occurs. This is manifested in the spectrum cures as the region where the wavenumber tends to infinite (or wave speed tends to zero). These results are also compared with the Born-Karman model and also with the second and fourth order strain gradient models. The results can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave propagation properties of single-walled carbon nanotubes.
Table of Contents
1. Introduction
2. Mathematical Formulation
2.1 A review on theory of nonlocal elasticity
2.2 Nonlocal strain gradient models
2.3 Nonlocal governing partial differential equation for nanorods basedon nonlocal stress gradient model
3. Ultrasonic Wave Characteristics Nanorods
3.1 Computation of wavenumbers
3.2 Escape frequency
3.3 Computation of wave speeds
4. Governing Partial Differential Equations of Nanorod
4.1 Based on second order strain gradient model
4.2 Based on fourth order strain gradient model
4.3 Discussion on second order strain gradient model
4.3.1Analytical solution
4.3.2Uniqueness
4.3.3Stability
5. Ultrasonic Wave Characteristics of Nanorod
5.1 For second order strain gradient model
5.1.1 Critical Wavenumber
5.1.2 Number of Waves Along the Nanorod
5.2 For fourth order strain gradient model
6. Dynamic Response of Nanorods
6.1 For second order strain gradient model
6.2 For fourth order strain gradient model
7. Nonlocal Governing Partial Differential Equation for Nanorods
8. Numerical Experiments, Results and Discussion
8.1 Stress gradeint model
8.2 Strain gradient models
8.3 Nonlocal stress gradient model with lateral inertia
9. Concluding Remarks
Research Objectives and Topics
This paper investigates the non-classical wave dynamics of ultrathin structures (nanorods) by incorporating nonlocal elasticity theory into classical rod models to capture small-length scale effects that influence wave propagation behavior at the nanoscale.
- Application of nonlocal elasticity theory to one-dimensional nanostructures.
- Development and comparison of nonlocal bar, second-order, and fourth-order strain gradient models.
- Analysis of wave characteristics, including dispersion, phase speed, group speed, and escape frequency.
- Investigation of lateral inertia effects and their role in stabilizing gradient models.
- Assessment of the dynamic response of nanorods under impact excitation using Laplace transforms.
Excerpt from the Book
2.1 A review on theory of nonlocal elasticity
According to the theory of nonlocal elasticity [13-15], the stress at a reference point x is considered to be a functional of the strain field at every point in the body. In the limit when the effects of strains at points other than x are neglected, one obtains local or classical theory of elasticity. The basic equations for linear, homogeneous, isotropic, nonlocal elastic solid with zero body force are given by:
σ ij,j = 0, σ ij(x) = ∫V α(|x-x'|, ξ)Cijklεkl(x')dV(x') ∀ x ∈ V, εij = 0.5(ui,j + uj,i), where Cijkl is the elastic modulus tensor of classical isotropic elasticity, σij and εij are stress and strain tensors respectively, and ui is the displacement vector.
α = α(|x-x'|, ξ) is the nonlocal modulus or attenuation function incorporating the nonlocal effects into the constitutive equations. This nonlocal modulus is found by matching the curves of plane waves with those due to atomic lattice dynamics. Various different forms of α(|x-x'|) have been reported in [13]. |x-x'| is the Euclidean distance, and ξ = e0a/l, where a is an internal characteristic length, e.g., length of C-C bond (0.142 nm) in CNT, granular distance etc., and l is an external characteristic length e.g., wavelength, crack length, size of the sample etc. e0 is a nonlocal scaling parameter,
Summary of Chapters
1. Introduction: Defines ultrathin structures and identifies the limitations of classical continuum mechanics when dealing with heterogeneous phenomena at the nanoscale.
2. Mathematical Formulation: Establishes the theoretical foundation, reviewing nonlocal elasticity and deriving constitutive equations for strain gradient models.
3. Ultrasonic Wave Characteristics Nanorods: Analyzes the dispersion relations, computes wavenumbers, and defines the escape frequency for nanorods.
4. Governing Partial Differential Equations of Nanorod: Derives governing differential equations for nanorods using energy methods for second and fourth order strain gradient models.
5. Ultrasonic Wave Characteristics of Nanorod: Investigates dispersion characteristics specifically for the second and fourth order strain gradient models and identifies critical wavenumbers.
6. Dynamic Response of Nanorods: Studies the dynamic impact response of semi-infinite nanorods using Laplace integral transforms.
7. Nonlocal Governing Partial Differential Equation for Nanorods: Introduces lateral inertia into the nonlocal governing equation to enhance the accuracy of the nanorod model.
8. Numerical Experiments, Results and Discussion: Presents and interprets simulation results, comparing different models against each other and classical/experimental baselines.
9. Concluding Remarks: Synthesizes findings, emphasizing the importance of nonlocal scaling and inertia parameters in nanodevice design.
Keywords
Nonlocal stress gradient model, Nonlocal strain gradient model, Lateral inertia, Wavenumber, Phase speed, Dispersion, Group speed, Nanorod, Born-Karman Model, Nanotechnology, Ultrathin structures, Carbon nanotubes.
Frequently Asked Questions
What is the core focus of this research paper?
The paper focuses on the non-classical wave dynamics of ultrathin structures, specifically nanorods, by integrating nonlocal elasticity theory into continuum mechanical models to account for nanoscale size effects.
What are the primary scientific fields covered in this study?
The work spans continuum mechanics, nanotechnology, wave propagation analysis, and material science, specifically regarding the modeling of carbon nanotubes.
What is the main objective of the proposed analytical models?
The objective is to capture the unique wave behaviors of nanorods—such as dispersion—that classical models ignore, providing a more accurate representation of materials at the nanoscale.
Which scientific methods are employed to derive the results?
The research uses nonlocal elasticity theory, Hamiltonian principles for energy minimization, and Laplace integral transforms to analyze wave dispersion and dynamic responses.
How does the second-order model compare to the fourth-order model?
While the second-order model is analytically simpler, the fourth-order model provides a better approximation of realistic dynamic responses and avoids some of the instability issues inherent in lower-order models.
What role does the nonlocal scaling parameter play?
The nonlocal scaling parameter is critical as it introduces band gaps in axial wave modes and directly influences the dispersion behavior and the frequency range at which waves can propagate.
Why is lateral inertia significant in this context?
Lateral inertia is crucial because it accounts for the transverse motion effects in one-dimensional models, helping to stabilize the second-order strain gradient model against non-physical instabilities.
What are the practical implications of this study?
The findings offer vital guidance for the design of next-generation nanodevices, such as those utilizing single-walled carbon nanotubes, by accurately predicting wave propagation characteristics.
- Citar trabajo
- S. Narendar (Autor), 2012, Non-classical Wave Dynamics of Ultrathin Structures, Múnich, GRIN Verlag, https://www.grin.com/document/201982
